Alternatively, by using the linearisation \(\mathcal {L}\)  to give an equivariant embedding \(Y \hookrightarrow \mathbb {P}^n\) , the Hilbert–Mumford criterion can be stated combinatorially using the weights of a maximal torus \(T < G\)  on \(\mathbb {P}^n\)  (or rather the affine cone over this projective space). A point is semistable for the \(T\) -action if its convex hull of \(T\) -weights contains the origin [1]}. Moreover, a point is semistable for the \(G\) -action if all points in its \(G\) -orbit are \(T\) -semistable; thus we have
\(Y^{G-ss} = \cap _{g \in G} \: gY^{T-ss}.\) 
